All Functions $g: \mathbb{N} \rightarrow \mathbb{N}$ Which have a Single-Fold Diophantine Representation are Dominated by a Limit-Computable Function $f: \mathbb{N}\setminus \{0\} \rightarrow \mathbb{N}$ Which is Implemented in MuPAD and Whose Computability is an Open Problem

Abstract

Let $E_{n} =\{ x_{k} = 1,\ x_{i} + x_{j} = x_{k},\ x_{i} \cdot x_{j} = x_{k}: i,j,k \in \{ 1,\ldots,n\}\}$. For any integer n ≥ 2214, we define a system $T \subseteq E_{n}$ which has a unique integer solution (a ₁, …, a _n ). We prove that the numbers a ₁, …, a _n are positive and $\mathrm{max}\left (a_{1},\ldots,a_{n}\right ) > 2^{2^{n} }$. For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system $S \subseteq E_{n}$ with a unique solution in non-negative integers x ₁, …, x _n , this solution belongs to [0, b]ⁿ. We prove that if a function $g: \mathbb{N} \rightarrow \mathbb{N}$ has a single-fold Diophantine representation, then f dominates g. We present a MuPAD code which takes as input a positive integer n, performs an infinite loop, returns a non-negative integer on each iteration, and returns f(n) on each sufficiently high iteration.